A002124 Number of compositions of n into a sum of odd primes.

1, 0, 0, 1, 0, 1, 1, 1, 2, 1, 3, 4, 3, 7, 7, 8, 14, 15, 21, 28, 33, 47, 58, 76, 103, 125, 169, 220, 277, 373, 476, 616, 810, 1037, 1361, 1763, 2279, 2984, 3846, 5006, 6521, 8428, 10983, 14249, 18480, 24048, 31178, 40520, 52635, 68281, 88765, 115211, 149593, 194381, 252280, 327696, 425587, 552527, 717721

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn

Comments

Arises in studying the Goldbach conjecture.

The g.f. -(z-1)*(z+1)*(z**2+z+1)*(z**2-z+1)/(1-z**6-z**3-z**5-z**7+z**9) conjectured by Simon Plouffe in his 1992 dissertation is wrong.

References

• N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• N. J. A. Sloane, Table of n, a(n) for n = 0..1000
• P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; see page 300
• P. A. MacMahon, Properties of prime numbers deduced from the calculus of symmetric functions, Proc. London Math. Soc., 23 (1923), 290-316. [Coll. Papers, II, pp. 354-382] [The sequence i_n]
• Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
• Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
• Index entries for sequences related to Goldbach conjecture

Crossrefs

Cf. A002125, A023360, A024939, A077608.

Cf. A065091.

Programs

Haskell

import Data.List (genericIndex)
a002124 n = genericIndex a002124_list n
a002124_list = 1 : f 1 [] a065091_list where
   f x qs ps'@(p:ps)
     | p <= x    = f x (p:qs) ps
     | otherwise = sum (map (a002124 . (x -)) qs) : f (x + 1) qs ps'
-- Reinhard Zumkeller, Mar 21 2014

Maple

A002124 := proc(n) coeff(series(1/(1-add(z^numtheory[ithprime](j),j=2..n)),z=0,n+1),z,n) end;
M:=120; a:=array(0..M); a[0]:=1; a[1]:=0; a[2]:=0; for n from 3 to M do t1:=0; for k from 2 to n do p := ithprime(k); if p <= n then t1 := t1 + a[n-p]; fi; od: a[n]:=t1; od: [seq(a[n],n=0..M)]; # N. J. A. Sloane, after MacMahon, Dec 03 2006; used in A002125

Mathematica

a[0] = 1; a[1] = a[2] = 0; a[n_] := a[n] = (s = 0; p = 3; While[p <= n, s = s + a[n-p]; p = NextPrime[p]]; s); a /@ Range[0, 58] (* Jean-François Alcover, Jun 28 2011, after P. A. MacMahon *)

Formula

a(0)=1, a(1)=a(2)=0; for n >= 3, a(n) = Sum_{ primes p with 3 <= p <= n} a(n-p). [MacMahon]

G.f.: 1/( 1 - Sum_{k>=2} x^A000040(k) ). [Joerg Arndt, Sep 30 2012]

Extensions

Better description and more terms from Philippe Flajolet, Nov 11 2002

Edited by N. J. A. Sloane, Dec 03 2006